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An ^th complex root (root of degree ) is one of the complex solutions of

Real roots

An ^th real root (root of degree ) is one of the real solutions of

If is an even positive integer, then the two real roots are

while if is an odd positive integer, then the single real root is

Surds

A surd is an algebraic irrational root, e.g. is a cubic surd. The quadratic surd is a mixed surd (i.e. a rational number multiplied by a surd).

See also

Hierarchical list of operations pertaining to numbers ^{[1] [2]}

0^th iteration

• Successor S(n)
• Predecessor P(n)

1^st iteration

• Addition, S(S(... s times ...(S(n)))), the sum n+s
• Subtraction, P(P(... s times ...(P(n)))), the difference n-s

2^nd iteration

• Multiplication, n+(n+(... m times ...(n+(n)))), the product n∗m
• Division, the ratio n/d
  • Quotient (integer division)
  • Remainder (modulo and congruences)

3^rd iteration

• Exponentiation (d as "dimension", b as "base", n as "variable")
  • Powers, n∗(n∗(... d times ...(n∗(n)))), written n^d
  • Exponentials, b∗(b∗(... n times ...(b∗(b)))), written b^n
    • Exponential function e^n, where e is Euler's number
• Exponentiation inverses
  • Roots,
  • Logarithms, log_bn
    • Natural logarithm function log n, or log_e, where e is Euler's number

4^th iteration

• Tetration (d as "dimension", b as "base", n as "variable")
  • Tetra-powers (super-powers), n^(n^(... d times ...(n^(n)))), written n^^d or n↑↑d
  • Tetra-exponentials (super-exponentials), b^(b^(... n times ...(b^(b)))), written b^^n or b↑↑n
• Tetration inverses
  • Tetra-roots (super-roots)
  • Tetra-logarithms (super-logarithms),
    • Iterated logarithm (log*),

5^th iteration

• Pentation (d as "dimension", b as "base", n as "variable")
  • Penta-powers, n^^(n^^(... d times ...(n^^(n^^(n))))), written n^^^d or n↑↑↑d
  • Penta-exponentials, b^^(b^^(... n times ...(b^^(b^^(b))))), written b^^^n or b↑↑↑n
• Pentation inverses
  • Penta-roots
  • Penta-logarithms

6^th iteration

• Hexation (d as "dimension", b as "base", n as "variable")
  • Hexa-powers, n^^^(n^^^(... d times ...(n^^^(n)))), written n^^^^d or n↑↑↑↑d
  • Hexa-exponentials, b^^^(b^^^(... n times ...(b^^^(b)))), written b^^^^n or b↑↑↑↑n
• Hexation inverses
  • Hexa-roots
  • Hexa-logarithms

7^th iteration

• Heptation (d as "dimension", b as "base", n as "variable")
  • Hepta-powers, n^^^^(n^^^^(... d times ...(n^^^^(n)))), written n^^^^^d or n↑↑↑↑↑d
  • Hepta-exponentials, b^^^^(b^^^^(... n times ...(b^^^^(b)))), written b^^^^^n or b↑↑↑↑↑n
• Heptation inverses
  • Hepta-roots
  • Hepta-logarithms

8^th iteration

• Octation (d as "dimension", b as "base", n as "variable")
  • Octa-powers, n^^^^^(n^^^^^(... d times ...(n^^^^^(n)))), written n^^^^^^d or n↑↑↑↑↑↑d
  • Octa-exponentials, b^^^^^(b^^^^^(... n times ...(b^^^^^(b)))), written b^^^^^^n or b↑↑↑↑↑↑n
• Octation inverses
  • Octa-roots
  • Octa-logarithms

Notes

1. ↑ Hyperoperation—Wikipedia.org.
2. ↑ Grzegorczyk hierarchy—Wikipedia.org.

Notes